Abstract

The sequence \(3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\ldots \) consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erdős–Ford–Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy–Ramanujan inequality.

Keywords

Gaussian primesPythagorean triples

Mathematics Subject Classification

Primary 11N25Secondary 11N0511N36

1 Background

The sequence OEIS A281505 concerns odd legs in right triangles with integer side lengths and prime hypotenuse. By the parametrisation of Pythagorean triples, these are positive integers of the form \(x^2 - y^2\), where \(x,y \in \mathbb N\) and \(x^2 + y^2\) is prime. Even legs are those of the form 2xy, where \(x, y \in \mathbb N\) and \(x^2 + y^2\) is an odd prime. Let \(\mathcal A\) be the set of odd legs, and \(\mathcal B\) the set of even legs that occur in such triangles. Consider the quantities

Theorem 1.1

Since every prime \(p\equiv 1\pmod 4\) is representable as \(a^2+b^2\) with a, b integral, we have \(\mathcal C(N)\) unbounded. In fact, using the maximal order of the divisor function, we have \(\mathcal C(N) \geqslant N^{1-o(1)}\) as \(N\rightarrow \infty \). We obtain a strengthening of this lower bound.

Theorem 1.2

Note that \(\log 4-1\approx 0.386\). Since \(\mathcal B(2N) = \mathcal C(N)\), we obtain the same bounds for \(\mathcal B(N)\). By essentially the same proofs, one can also deduce these bounds for \(\mathcal A(N)\).

To motivate the outcome, consider the following heuristic. There are typically \(\approx (\log n)^{\log 2}\) divisors of n, which follows from the normal number of prime factors of n, a result of Hardy and Ramanujan [8]. Moreover, given a factorisation \(n=ab\), the “probability” of \(a^2+b^2\) being prime is roughly \((\log n)^{-1}\). Since \(\log 2 < 1\), we expect the proportion to decay logarithmically. In the presence of biases and competing heuristics, this prima facie prediction should be taken with a few grains of salt. We use Brun’s sieve and the Hardy–Ramanujan inequality to formally establish our bounds. In addition, for Theorem 1.2 we use a result of Harman and Lewis [9] on the distribution of Gaussian primes in narrow sectors of the complex plane.

We write for the set of primes. We use Vinogradov and Bachmann–Landau notation. As usual, we write for the number of distinct prime divisors of n, and for the number of prime divisors of n counted with multiplicity. The symbols p and \(\ell \) are reserved for primes, and N denotes a large positive real number.

2 An upper bound

In this section, we establish Theorem 1.1. The Hardy–Ramanujan inequality [8] states that there exists a positive constant \(c_0\) such that uniformly for \(i \in \mathbb N\) and \(N\geqslant 3\) we have

Remark 2.1

Note that we might equally well have used the version of Brun’s sieve from [7, p. 68], which is less precise, but somewhat easier to utilise. In fact, as kindly suggested by one of the referees, one could accomplish the same result using Brun’s pure sieve [6, Eq. (6.1)], which is nothing more than a strategic truncation of the inclusion-exclusion principle.

Remark 3.2

The problem of counting Gaussian primes in narrow sectors has received quite some attention over the years, and still it is far from resolved. Rather than using Theorem 3.1 by Harman and Lewis [9], we could have used a weaker result by Kubilius [10] from the 1950s. We refer the interested reader to the introduction of [2] for more about the earlier history of this problem.

The fact that \(u \ne v\) ensures that there are three primality conditions defining \(S(g,u,v,w_0)\). To bound \(S(g,u,v,w_0)\) from above, we may assume without loss that \(guvw_0\) is even, and that the variables \(g,u,v,w_0\) are pairwise coprime, for otherwise \(S(g,u,v,w_0) = 0\). Paralleling Sect. 2, an application of Brun’s sieve reveals that

4 A final comment

We conjecture that Theorem 1.1 holds with equality. For a lower bound, one might restrict attention to those pairs (a, b) with \(\omega (a)\approx \omega (b)\approx \frac{1}{2\log 2}\log \log N\). The upper bound for the second moment is analysed as in the paper, getting \(N/(\log N)^{\eta +o(1)}\); we expect that a more refined analysis would give

here. The difficulty is in obtaining this same estimate as a lower bound for the first moment. This would follow if we had an analogue of Theorem 3.1 in which a, b have a restricted number of prime factors. Such a result holds for the general distribution of Gaussian primes, at least if one restricts only one of a, b, see [5].

Declarations

Author's contributions

SC and CP jointly proved the theorems, drafted the manuscript, and polished it. Both authors have read and approved the final manuscript

Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Acknowledgements

The authors were supported by the National Science Foundation under Grant No. DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. The authors thank John Friedlander, Roger Heath-Brown, Zeev Rudnick, Andrzej Schinzel and the anonymous referees for helpful comments, and Tomasz Ordowski for suggesting the problem.

Dedication

This year (2017) is the 100th anniversary of the publication of the paper On the normal number of prime factors of a number n, by Hardy and Ramanujan, see [8]. Though not presented in such terms, their paper ushered in the subject of probabilistic number theory. Simpler proofs have been found, but the original proof contains a very useful inequality, one which we are happy to use yet again. We dedicate this note to that seminal paper.

Competing interests

The authors declare that they have no competing interests.

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